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1 Measuring evolution of systemic risk across insurance-reinsurance company networks Abstract Constructing a stochastic model to quantify counterparty risk in the insurance industry Create estimators based on the model to measure key statistics such as Ruin Probability and LGD Improves upon the Eisenberg-Noe framework by netting defaults in a set of insurance companies Authors: James Ma, Juan Li, Jose Blanchet, Yixi Shi

3 Risk propagates throughout the network via interconnected arrows l i : Insurance company i R i : Reinsurance company I U i : Risk factor i (claims)

4 Insurance-Reinsurance Model Parameters Insurance companies reside in the set {I} Reinsurance companies reside in the set {R} Reinsurance companies are never insurance companies. Opposite is also true. An insurance company enters contract with one or more reinsurance companies. The reinsurers bear some proportion of the losses taken by an insurer. Li, Lr are net immediate losses born by the insurer i and reinsurer r Fixed premiums of Ci, and Cr are paid by the insurer and the reinsurer each period Initial endowment (aka reserves) of e i and e r for insurance i and reinsurance r Spill-over proportions of loss at the point of default are ρr,i and ρi,r

5 What we want to measure Ruin Probability: Ruin probabilities of a particular set of insurance companies LGD: Loss given default in the system given the default of a set of companies and taking into account spill-over effects Impact of Reinsurance: How do reinsurance companies affect the risk of the interconnected system

6 Default-Contagion Mechanism Graphical Illustration A claim of 50 enters the network. I1 is able to recover 20 from R2. It retains 30 in loss. Because reinsurance R1 has a loss proportion of 0.5 with R2, R2 passes 50%, or 10, of its loss to R1. At the other end, a claim of 80 enters the network into I2. I2 passes 40 in loss to R1, because of the 40 in deductible from 80. I2 retains 40 losses. R1 passes 20 of its losses assumed from I2 to R2, due to 0.5 risk sharing agreement. R1 thus retains 10 from R2 and 20 from I1, for a net total retained loss of 30.

7 Default Mechanism From this LP, one can estimate that given a default of a set of companies, the optimal loss minimizing mechanism of premiums and recoverables Improves upon Eisenberg-Noe By netting defaults

8 Risk Factor Model A set of risk factors U are modelled by a matrix of m independent power-law-tail distribution, where b is the constant amount that the claims can exceed αj will be estimated from real-world data of multi-year claim trend

9 Blanchet and Shi Ruin Probability Theorem The theorem derives a relationship between the ruin probability of a set of companies and the loss amount. Gauges the ruin probability of the set using this theorem

10 Role of Reinsurance We now integrate the reinsurance companies into the ruin probability model derived earlier. e r represents the initial endowment of the reinsurance companies.

11 Model Answers to Regulatory Questions The ruin probability model including reinsurance companies helps to answer questions such as: Minimum capital requirement (e) for the reinsurers with a specific level of failure probabilities What is the most vulnerable set of insurance companies within the system by gauging capital requirements for a given level of failure rate

12 Monte Carlo Simulation In order to apply the theoretical reinsurance model to real-world data, we need to construct a Monte Carlo estimator of the ruin probability framework The variance of the estimator will decrease as b converges to zero, thus making the model a robust measure of ruin probability given a rare-event

13 Monte Carlo Algorithm Using this sampling algorithm combined with real world data, we can estimate a robust Z A such that can be used to construct an optimal risksharing mechanism within an insurancereinsurance network. Z A is the estimator for the ruin probability of the insurancereinsurance set.

14 Numerical Example Given three hypothetical scenarios of risk factor parameter α from each of the Z insurance-reinsurance companies in the set, we can combine the parameters with the LP data (a separate matrix of data including losses, premium, and recoverable) to arrive robust estimation of the ruin probability and the LGD

15 Numerical Example Results

16 Application to Travelers Network in Progress Matrix above used to estimate the LP to arrive at the optimal lambda, aks sample of x s Direct and assumed losses from multiyear business accounting used to estimate alpha from the Risk Factor distribution

17 Conclusion and significance Advantages of the Blanchet-Shi Model Equilibrium settlements and netting effects Heavy-tailed factors useful for rare-event simulation Role of reinsurance in a risk network Bounded relative error

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